A Definition of Metastability for Markov Processes with Detailed Balance

TL;DR

This paper proposes a formal definition of metastable states in finite Markov processes with detailed balance, emphasizing a key probability condition, and demonstrates its application on a 2D kinetic Ising model.

Contribution

It introduces a new criterion for identifying genuine metastable states in Markov processes with detailed balance, linking physical intuition with mathematical formalism.

Findings

Identified a key condition for genuine metastability: negligible equilibrium probability in the state.

Applied the formalism to a 2D kinetic Ising model with preliminary results.

Connected the definition to the restricted ensemble approach.

Abstract

A definition of metastable states applicable to arbitrary finite state Markov processes satisfying detailed balance is discussed. In particular, we identify a crucial condition that distinguishes genuine metastable states from other types of slowly decaying modes and which leads to properties similar to those postulated in the restricted ensemble approach \cite{pen71}. The intuitive physical meaning of this condition is simply that the total equilibrium probability of finding the system in the metastable state is negligible. As a concrete application of our formalism we present preliminary results on a 2D kinetic Ising model.